Talk:Taylor series/Archive 3

Archive 1

Archive 2

Archive 3

Definition of series

From Wikipedia: https://en.wikipedia.org/wiki/Power_series In mathematics, a power series (in one variable) is an infinite series of the form

$\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+\cdots$

End quote. The formula and definition of series already contains the summation to infinity which hopefully converges.

A series Power/Taylor is informally speaking an infinite polynomial where the infinite summation converges desirably and whithin the convergence radius. It's not the sequence

$a_{0},\ \ a_{1}(x-c)^{1},\ \ a_{2}(x-c)^{2},\cdots$

which is summed up to polynomials which converge. Therefore a function can be equal to its Taylor series. One doesn't need to say sum of its Taylor series. See also the wording here (same as former version of the article): https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-5-exploring-the-infinite/part-b-taylor-series/session-98-taylors-series/MIT18_01SCF10_ex98sol.pdf

From this article: The Taylor series for any polynomial is the polynomial itself.

End Quote, If this is true (which I think too), then the Taylor series is not!!! the sequence,

$a_{0},\ \ a_{1}(x-c)^{1},\ \ a_{2}(x-c)^{2},\cdots$

but summation has already occured even of oo-many terms. (LMSchmitt 12:02, 24 November 2020 (UTC))

Obviously due to the latter citation from THIS article, the TS is also not a procedure.

Edit will be reverted. PS: will add another ref soon.

LMSchmitt 15:00, 24 November 2020 (UTC)

Before going any further, can you please clarify: would you understand what I meant if I were to say "This argument is about formal power series, and does not involve notions of convergence" about a certain computation? --JBL (talk) 17:54, 24 November 2020 (UTC)

JayBeeEll [A} If you have anything of value to contribute such as the valuable points of D.Lazard below, then please feel free to do so. Otherwise, I would be very appreciative, if I don't have to read contributions as the above in the future. [B] If you want to state that "This argument is about formal power series, and does not involve notions of convergence", then please just do so. I would not concur. [C] If one says "A function f equals its Taylor series." (**) [as is done by myself, and others at MIT (as documented in the above quoted online article/teaching document)], then that implies [1] one evaluates the (if you so desire) formal Taylor series by the usual procedure at any arbitrary point x in the domain of definition of f, and [2] the partial sums (usual procedure) must converge to a number y(x), and y(x)=f(x) must hold. [1]-[3] are quite obviously implicitly contained in the meaning of the latter sentence (**). Since you are mathematically educated, this should be obvious to you, or at least obvious from the referenced document. --- This discussion is about saying "A function equals its Taylor series." which implies convergence of a formal power series (if you like), but is not primarily about formal power/Taylor type series, but rather about their sum/limit/value coinciding with a function value. $\ \ \$LMSchmitt 00:29, 25 November 2020 (UTC)

Since I am one of the instructors of the online course whose notes you keep referencing, you will pardon me if I suggest that your condescending tone is badly misplaced. Indeed, you have succeeded in convincing me that trying to discuss anything with you will be deeply unpleasant, so I will now excuse myself from this conversation. Separately, I have fixed your incompetent errors below that damaged D.Lazard's comment, as well as the accompanying mis-indentation. --JBL (talk) 00:53, 25 November 2020 (UTC)

For the record, I understood your initial question as condescending as well, and unpleasant at the same time. LMSchmitt 02:57, 25 November 2020 (UTC)

The wording "incompetent errors" borders abusive language. That's not the usual style here. LMSchmitt 02:57, 25 November 2020 (UTC)

For the record, my "errors" are misfunctions of the Wikipedia software, or my browser (unlikely). Sometimes, while typing into the WP interface, whole paragraphs are duplicated somewhere below. That must have happened while I was typing my reply to you. Unfortunately, I did not notice it, because I concentrated on getting my typing right, and rechecked that often while assuming everything else untouched. LMSchmitt 02:57, 25 November 2020 (UTC)

It is not standard, and violates WP rules to edit other's talk page contributions. Next time, please notify me on my talk page. --- But, thank you for your corrections this time. Overall, they were useful. LMSchmitt 03:49, 25 November 2020 (UTC)

(edit conflict) In the article, this editor has twice systematically changed phrases like "the function may differ form the sum of its Taylor series" into "the function may differ form its Taylor series". This is a major modification because, if accepted, it woud imply to rewrite completely the lead of Series (mathematics), and to modify many Wikipedia articles. So, per WP:BRD, a consensus must be reach here before such a change. I explain below why this change would make the terminology inconsistent.

Thus I have restored the previous stable version. Please do not reinstate your own version without a consensus here.

Mathematical comment: It is classical that an addition is an operation, that is a computation or a processus, while a sum is the result of this operation. A witness of this basic distinction is the sentence "the sum of two positive numbers is greater than either" quoted from Addition: nobody will understand the sentence "the addition of two positive numbers is greater than either". This basic but rather subtle distinction is the same for series, and is clearly established in the lead of the article Series (mathematics). If a series were the limit of its partial sums, divergent series could not exist because the limit does not exist. This distinction between a series and its sum cannot be reflected in formulas; this makes irrelevant the above quotation of Power series.

So, given a sequence $(a_{1},a_{2},a_{3},\ldots ),$ "a series is the operation of adding the a_i one after the other" (quoted from Series (mathematics)), and the sum of a series is the result (if it exists) of this infinite sequence of operations. When working with convergent series, it is common to not emphasize on this distinction when this simplify wordings. Similarly, when it is not important, one does not distinguish between an addition and the sum which is the result of this addition.

On the other hand, identifying systematically series with their sum would makes the theory inconsistent, because one could not talk of a divergent series. This is what is implied implicily by the edits of LMSchmitt, and explicitly by their above post. D.Lazard (talk) 18:29, 24 November 2020 (UTC)

D.Lazard Thank you for your effort. We can keep "your" version. I do not agree with several of your points, but overall, I agree with your choice of standardized language. Where I come from, power/Taylor series means the usual formal infinite sum as displayed above, or simply

$\sum _{0}^{\infty }\star \equiv \lim _{N\to \infty }\sum _{0}^{N}\star$

\i.e., a shortcut for a limit notation and that is a clear definition. (It yields the same math in regard to analysis.) The limit can exist or not (then it's named divergent=non-convergent series) at a point. There is no inconsistency. If one says as others do (see reference above) "A function equals its Taylor series." Then this means: evaluate the function (say for all x), evaluate the series by establishing a limit of the partial sums at x (above limit notation), and both values coincide. As German Wikipedia points out "Reihe" (series) is most often understood/identified with its value (limit,sum), if the latter exists, REF: German Wikipedia, Reihe, 5 Semantik. --- Note that I never intended or proposed to identify every series with its possible sum/value/limit. $\ \ \$LMSchmitt 00:29, 25 November 2020 (UTC)

Madhava of Sanfamagrama

Actually, I think Archimedes should be accredited with the first use of the Taylor series, since he used the same method as Madhava: using an infinite summation to achieve a finite trigonometric result. Liu Hui independently employed a similar method 400 years later, but still about 800 years prior to Madhava's work, although the Wikipedia article on Liu Hui does not reflect this.

In fact, it would have been quite easy for them to perform the same task as Madhava. It isn't difficult to square an arc (albeit in an infinite number of steps) using simple Euclidean geometry. I believe that Archimedes and later Liu Hui were aware of this. Last time I heard about it was at a History and Philosophy of Mathematics conference in 1998 at the Center for Philosphy of Science, University of Pittsburgh. Anyone care to dredge up a reference? 151.204.6.171 — Preceding unsigned comment added by 151.204.6.171 (talk) 21:37, 3 November 2005 (UTC)

Taylor series with Lagrange and Peano remainders

Why there's nothing about those two remainders in the article? — Preceding unsigned comment added by 87.206.246.136 (talk) 14:35, 7 January 2007 (UTC)

New external link

I placed a new external link with an interactive demonstration of the Taylor series in 1D and 2D. Feel free to check if it is appropriate for this article. The link is this one :

• Interactive demonstration of Taylor Series for 1D and 2D functions

Allan Martins (talk) 01:22, 2 April 2021 (UTC)

It is definitively not appropriate for Wikipedia, as this is original research. See WP:No original research and WP:ELNO. D.Lazard (talk) 07:57, 2 April 2021 (UTC)

Indeed, as was explained extensively at User talk:DVdm#Reversion on the Taylor Series page. - DVdm (talk) 09:07, 2 April 2021 (UTC)

The APP is NOT original research. So according to WP:No original research and WP:ELNO, Im reverting your edit and understand that you did an unwilling mistake for not being a technical person on the subject. No problems. So, since you did a BOLD revert wp:BOLD, revert, discuss cycle by mistake, I'm editing it again. Please don't revert it. The like is in total accordance with Wikipedia rules and insisting on reverting it does not agree with Wikipedia good manners.

This application is original research in the sense of Wikipedia. Moreover, for being acceptable, you must provide WP:Reliable sources that discuss it and allow establishing its WP:Notability. Please stop WP:edit warring, and be care that continuing this way you may be blocked for editing per WP:3R rule. D.Lazard (talk) 11:24, 2 April 2021 (UTC)

This application is NOT original research in the sense of Wikipedia. Your opinion and interpretation of the rules need to be a consensus to be applicable wp:CONSENSUS. So, before reverting it, please try to discuss and reach a consensus. None of the links provide reliable source because they are external, please consult WP:Reliable sources. Im not reverting anything, I'm editing the article. You are reverting it and did two reverts in sequence, please be advise that you you revert my edit again you may be blocked for editing per WP:3R rule. I must say it again and again that you patrol work is important and very valuable for wikipedia. We are thankful for it. Please notice that in this case you are OVER DOING it. You have no technical background on the matter and is trying to revert an edit based on your opinions this is not ok WP:New_pages_patrol. Allan Martins (talk) 11:56, 2 April 2021 (UTC)

You need to stop trying to add this, per wp:NOCONSENSUS and wp:ELNO. You are few inches away from a block. Formal warnings on your user talk page. - DVdm (talk) 12:24, 2 April 2021 (UTC)

I am not reverting anymore because you all are engaging in edit-war with me based on you personal opinions. It's a pity that Wikipedia have such bad patrol which reverts content inappropriately and refuses to discuss a consensus. I'm willing to discuss and reach a consensus. But, since you are abusing some kind of authority and using threats of blocking as arguments (instead to pacifically discuss and reach a consensus) I have no option other to submit to your abuse of authority and leave. Allan Martins (talk) 12:41, 2 April 2021 (UTC)

I concur with Allan Martins (talk). I find the threat by DVdm (talk) above impolite, bullying, unacceptable behavior. I also find the insistance of D.Lazard (talk) on only his interpretation of rules utterly reprehensible. I can quote revert-edits of D.Lazard where a better "false" wording of a novice was reverted to the "right" wording, but when I asked two native speakers of English what the meaning of the "false" English is, it was the same as D.Lazard's explanation (which he kindly provided), i.e., the novice was right and his edit wasn't questionable at all. Poeple can make mistakes with edits and reverts. But it is bad when it's always the same pattern. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 05:07, 4 April 2021 (UTC)

Please sign all your talk page messages with four tildes (~~~~) — See Help:Using talk pages. Thanks.

The "threat" above was preceded by a chat at User talk:DVdm#Reversion on the Taylor Series page, the chat here, and multiple formal warnings at User talk:Ninguem wiki. And it was the not the first time that this user had tried to add their personal work to the external links section. We have wp:ELNO for a reason.

And that still does not establish a consensus to add another external link to a list that was too long to begin with. - DVdm (talk) 08:53, 4 April 2021 (UTC)

DVdm (talk). Thanks for your reply. Appreciated. In my opinion, Wikipedia has degenerated to a competition of rule-fetishists. Discussions are not about reason and reasonable content, but who can throw around the most rules in the current discussion to silence opposing opinions. In my opinion, it is a perversion, and I am so appaled in regard to WP that I don't contribute anymore. ---- Also, please, dont kid yourself, WP-math contributions are full of proofs which are not referenced and definitely look like someone proudly did his "homework." Thanks for your reply again. LMSchmitt 09:16, 4 April 2021 (UTC)

Please indent your talk page messages as outlined in wp:THREAD and wp:INDENT — See Help:Using talk pages. Thanks.

Yep, other homework exists. Most of it likely survived by consensus, or by obscurity. Let's at least try not to create even more... - DVdm (talk) 10:00, 4 April 2021 (UTC)

Definition via Taylor Series

The statement that transcendental function e.g., Exp(x), are defined by differential equations (DEs) is fundamentally flawed and should be removed. If one has a DE such as f'=f, f(0)=1, then one needs an existence theorem which should show existence and uniqueness. Just "abracadabra, hush/hush, puff" we have a function that we need to solve our DE is not a proper method. Math is not make-belief, and the current wording suggests that one can simple do that in math which gives a bad impression to the novice reader. See Rudin's book "Principles of Modern Analysis" for the def of Exp. [Of course, one can use f'=f as motivation to derive the respective Taylor/Power series that defines Exp.]

(a) In school, the exponential function is defined as inverse of log, and Exp'=Exp emerges naturally via reverse engineering the definition.

(b) The definition via the Taylor/Power series is much better and should be noted/emphasized as method of definition for transcendental functions. It is quite easy to use elemental estimates to show that it converges and is well-defined, but it is intuitively well-understandable, since readers understand Taylor-polynomials from school.

(c) Defining Exp via its Power series generalizes with the same proof of existence and convergence to matrices or bounded linear operators.

LMSchmitt 11:23, 23 November 2020 (UTC)

Are you proposing something? Because right now the discussion of defining exp as the solution of a certain differential equation is a very nice motivational piece of text, and is accurate without burdening the reader with unnecessary technicalities. --JBL (talk) 13:16, 23 November 2020 (UTC)

Are you not educated enough to see that I am proposing something.? I am annoyed that instead of presenting solid sound reason, you request to answer a submissive question before you present your not-so-solid argument. In fact, you follow the "weakly" ad hominem procedure: (1) ask an unnecessary, innocent question, indirectly attacking the "opponent" (your questions are not insulting, thus "weak ad hominem"), then present a "popular opinion" instead of presenting solid argument. Without further (strong?) theorems defining exp via a diff eq is just bull. LMSchmitt 08:13, 6 April 2021 (UTC)

The definition by a power series does not explain why one chooses these specific coefficients. On the other hand, the definition by a differential equation explains the importance of the exponential function, because one cannot imagine a simpler differential equation than $y'=y.$ Also, in this case the radius of convergence is infinite, and thus the Taylor series define the exponential everywhere. But, for most transcendental functions this is not the case. For example, the natural logarithm is commonly defined by the differential equation $xy'=1$ (that is to say that the log is an antiderivative of $1/x$). On the other hand, the Taylor series at 1 has a radius of convergence of 1, and thus does not define $\log 10.$ The same is true for almost all usual transcendental functions. Moreover, using the Taylor series for defining a function amounts to replace a differential equation by the recurrence relation satisfied by the coefficients, which is rarely a simplification, although this is almost equivalent (see Holonomic function). D.Lazard (talk) 14:09, 23 November 2020 (UTC)

This again shows again that you prefer bullshitting people over solid math arguments just to be "right." In fact, you are pathetically wrong. [1] S.Langs book "Analysis I" beautifully presents the differential equation f=df and deduces elemental properties of exp from it, BUT postpones the proof of existence of exp until the chapter about power series. W.Rudin in his books defines exp via power series. [2] I aknowledge and I do know, that using f=df yields a nice HEURISTICS what exp is good for, and what the power series of exp must look like (as you point out), BUT it does not prove by any means that such a nice function exists. Claiming (as you do) that saying exp is the solution to f=df and PRETENDING that this is enough of definition is intellectual fraud (on your part and in general). Despite the nice heuristics, one needs the "school math" definition, or the power-series definition or another one to show that exp exists. YOU ARE FUCKING BLOODY WRONG in arguing that f=df defines exp (in the sense of existence). [2] No-one but you on this planet defines LOG as solution of xy'=1. In any mathbook I have seen in my life LOG is exp^{-1} or log=int{1/x}. It is more than pathetic that you desparately cling to "defining certain functions via diff eqs." Unless, you invoke a general theorem about existence of solutions of types of diff eqs, a diff eq yields nice+good+valuable heuristics, but BY FAR doesnt yield existence. The text in this article is just BULL. LMSchmitt 08:13, 6 April 2021 (UTC)

Formula Expansion for Several Variables

In the section for several variables, the lines expanding

$T(x_{1},\ldots ,x_{d})=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})$

divide all terms of the form

$(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})$ where $n_{1}+\cdots +n_{d}=z$

by the same $z!$.

Is that correct? For example, shouldn't terms of the form ${\frac {\partial ^{2}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})$ where $j\neq k$ be divided by $1!1!$ instead of $2!$?

D4nn0v (talk) 04:56, 1 August 2021 (UTC)

Both $j=1,k=2$ and $j=2,k=1$ contribute to that term so dividing by 2 is necessary. I didn't check the general case; let us know if you still think it is wrong. McKay (talk) 07:46, 5 August 2021 (UTC)

This is a perennial point of confusion, see Talk:Taylor_series/Archive_2#Multi-index_notation and assorted other discussions in the archives of this talk-page. --JBL (talk) 14:05, 5 August 2021 (UTC)

Makes sense now. Thanks McKay and JBL. --D4nn0v (talk) 08:34, 6 August 2021 (UTC)